Optimization problem that (very) loosely parallels

[Overseer55]

I came up with this “game” yesterday evening after reading (on 2+2) a post from a well respected individual that “it is possible to poker perfectly, but the definition of ‘perfectly’ changes all the time”.

I find the "game" to be fairly interesting. Also, I think it is a reasonable model for optimizing "perfect play" in a game where the definition of "perfect play" is constantly changing.

I call it the "triple-half" game.

Specifications:
Let game G consist of 100 trials. At the beginning of each trial, you are given a $1000 “trial balance” and are guaranteed that the trial is “fair”. Each trial consists of a potentially unlimited number of rounds. At each round, you are allowed to move your “trial balance” to your “overall balance” or you can risk the “trial balance” in the hopes of winning more. If you decide to risk the money, two events happen (in succession). First, a random number, n, between 0 and 1 is chosen. If n is less than p, the trial becomes “unfair”. Second, a coin is flipped and your trial balance changes. If the result is heads, your trial balance triples. If the result is tails, your trial balance is reduced by 50%. If the trial is in “fair” mode, the coin has an equal chance of landing on heads and tails. If the trial is in “unfair” mode, the coin will always land on tails.

Goal:
Determine an optimal strategy to maximize your overall balance.

Clarifications:
1) Once a given trial becomes “unfair”, there is no way to make it “fair” again.
2) You do not know when a given trial becomes “unfair”.
3) You do not know p at the beginning of the game.

If my analysis is correct,

1) It is possible to optimally play the game if you know p (using some form of Bayesian analysis).
2) Every trial allows you to refine your estimation of p.
3) By playing more rounds in a trial that you suspect to be unfair (not necessarily the first one) you increase your ability to estimate p.
4) Assuming that 'p' is uniformally distributed from [0,1) at the beginning of the game may not be optimal.

I may be out in left-field with this one...but, hopefully someone else in the world will find this one interesting. BTW, I have taken a significant number of university level math/probability/statistics courses...so, feel free to throw any formulas at me.

Thanks in advance.

- Mark